WX UMa's radio spectrum shows a rapid decline within the observing band which could be a result of beaming of cyclotron emission or due to cyclotron absorption of plasma emission. Alternatively, the radio emission we observe might have originated not on WX UMa but on a sub-stellar companion. All three possibilities can be tested with longer term multi-frequency monitoring of WX UMa's radio emission.

ID completed this work as part of the ASTRON summer student programme. JRC thanks the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for support via the Talent Programme Veni grant.
AAV and TPR acknowledge funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 Research and Innovation Programme
(grant agreements No 817540, ASTROFLOW and No 743029 EASY respectively).
AD acknowledges support by the BMBF Verbundforschung under the grant 05A20STA.
This paper is based on data obtained with the International LOFAR Telescope (obs. IDs 233996, 441476, 403966 and 403968) as part of the LoTSS survey. LOFAR is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefitted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d'Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK.
This research made use of the Dutch national e-infrastructure with support of the SURF Cooperative (e-infra 180169) and the LOFAR e-infra group. The Jülich LOFAR Long Term Archive and the German LOFAR network are both coordinated and operated by the Jülich Supercomputing Centre (JSC), and computing resources on the supercomputer JUWELS at JSC were provided by the Gauss Centre for supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for Computing (NIC).
This research made use of the University of Hertfordshire high-performance computing facility and the LOFAR-UK computing facility located at the University of Hertfordshire and supported by STFC [ST/P000096/1], and of the Italian LOFAR IT computing infrastructure supported and operated by INAF, and by the Physics Department of Turin university (under an agreement with Consorzio Interuniversitario per la Fisica Spaziale) at the C3S Supercomputing Centre, Italy.
The Jülich LOFAR Long Term Archive and the German LOFAR network are both coordinated and operated by the Jülich Supercomputing Centre (JSC), and computing resources on the supercomputer JUWELS at JSC were provided by the Gauss Centre for Supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for Computing (NIC).
Radio data processing
The data were processed with the usual LoTSS pipelineThe LoTSS pipeline can be found at https://github.com/mhardcastle/ddf-pipeline [1]}, [2]} which is described in [3]}. We then performed an additional round of self-calibration in the direction of WX UMa with a procedure described in [4]} and also employed in [5]}.
After calibration, the measurement set was further processed to make source extraction easier for producing light curves and spectra. WX UMa was removed from the mask image which was then used to produce a model file using WSClean. WX UMa was also removed from these model images so that it would not be removed during the subtraction of other sources. The subtracted model image was then used with WSClean again to predict visibilities which were recorded in a new modelled data column in the measurement set. The modelled data was subtracted from the real data column and recorded in a new column - this served to remove all other bright sources from the region besides WX UMa. Finally, DPPP was used to phaseshift the measurement sets to be centred on WX UMa, using the subtracted data as the new real data.
After extra sources were removed from the field and the set was phaseshifted, frequency and time slices for Stokes-I and V were produced via WSClean. Because the signal to noise ratio of the observation was not large enough to produce a full dynamic spectrum, the frequency slices were averaged over all time and the time slices were averaged over the first 40 channels (120 to to 135.83 MHz, \(\approx \) 16 MHz bandwidth) where WX UMa is the brightest. The Background and Noise Estimation (BANE) tool in Aegean [6]} was used with each image to produce rms and background images. Aegean was run with a 4\(\sigma \)  seedclip to avoid identifying extraneous sources. The flux density for WX UMa for a particular image was assigned from the peak flux found by Aegean if a source was found in the expected location of WX UMa, and the rms noise was retrieved from the rms image produced by BANE. For images where no 4\(\,\sigma \)  source was found at the star's coordinates, a priorized fitting was applied using the catalogue from the image with the most significant detection in order to extract \(\approx 3\,\sigma \)  detections at the expected position of WX UMa. The images were also inspected by eye to make sure the results of Aegean were consistent with the images. The temporal evolution of WX UMa's radio flux is shown in Fig. REF  and its spectral evolution is shown in Fig. REF .
We note that the some the images in the light curve estimation have non-Gaussian noise due to residual sidelobe noise (from poorer \(uv\)  coverage) and low-level calibration errors.
Stokes-V sign convention
There are two commonly used Stokes-V sign conventions in literature. The pulsar literature typically follows the convention \({\rm V} = {\rm LCP}- {\rm RCP}\)  whereas the IAU convention is \(V = {\rm RCP} - {\rm LCP}\)  [7]}. To check the convention applicable to the LoTSS images, we compared the sign of Stokes-V of several pulsar detections to that in the literature and found that the LoTSS survey data products follow the same convention as the one in pulsar literature [8]}. Hence, WX UMa is radiating \({\rm RCP}\)  radiation, which means that the electric field vector rotates counter-clockwise as a function of time to an observer looking at incoming radiation.

Emission mechanism
Brightness temperature of plasma emission
In Fig. REF , we computed the brightness temperature of plasma emission using the expressions of [9]}. In doing so, we assumed a peak turbulence level of \(w=10^{-5}\)  and to compute the plasma temperature, we used the empirical relationship of [10]} and the observed X-ray luminosity (Table REF ).
<FIGURE>
Escape of plasma emission
Fundamental plasma emission is excited at the upper-hybrid frequency: \(\nu _{\rm uh} \approx \sqrt{\nu _p^2+\nu _c^2}\)  where \(\nu _p\)  and \(\nu _c\)  are the plasma and cyclotron frequencies respectively. Note that the upper-hybrid frequency is always larger than the cyclotron frequency.
The cyclotron absorption optical depth for \(o\) -mode radiation propagating at an angle \(\theta \)  to the magnetic field may be approximated by
\(\tau = \left(\frac{\pi }{2}\right)^{5/2}\frac{2}{c}\frac{\nu _p^2}{\nu }\frac{s^{2s}}{s!}\left(\frac{k_BT\sin ^2\theta }{2m_ec^2} \right)^{s-1} (1+|\cos \theta |)^2 L_B\) 
where integer \(s=\nu /\nu _c>1\)  is the cyclotron harmonic number, \(T\)  is the coronal temperature, \(m_e\)  is the electron mass, \(c\)  is the speed of light, \(k_B\)  is Boltzmann's constant and \(L_B\)  is the magnetic scale length which we take to be \(R_\ast \) . For the parameters of interest: \(T\sim 10^{6.7}\,{\rm K}\) , \(L_B\sim 10^{10}\,{\rm cm}\) , the optical depth is safely below unity only for \(s\ge 4\) . Hence, the radiation escape condition is \(\sqrt{\nu _p^2+\nu _c^2}> 4\nu _c\) , which gives \(\nu _c<0.26\nu _p\) .

Spectral evolution of the emission-cone
Consider a single dipolar field line which intersects the equatorial plane at radial distance \(L\) . The radial distance and polar angle, \(\theta ^\prime \)  along the field line are related via \(L = R\sin ^2\theta ^\prime \) . The polar inclination angle of the field lines, \(\alpha \)  is given by \(\cos \alpha = (3\cos ^2\theta ^\prime -1)(1+3\cos ^2\theta ^\prime )^{-1/2}\) . The magnetic field strength (and cyclotron frequency) vary continuously along the field line according to \(\nu (R) = \nu _0R^3_\ast /(2R^3)(4-3R/L)^{1/2}\) , where \(\nu _0\)  is the peak cyclotron frequency at \(\theta ^\prime =0,\,R=R_\ast \) . With the above relationships, we predicted the inclination of the emission cone axis for various \(L\) -shell at a range of cyclotron frequencies (see Fig. REF ). The numerically computed slope of the \(\alpha \)  versus \(\nu \)  lines at \(140\,{\rm MHz}\)  vary between \(\partial \alpha /\partial \nu \approx 0.2^\circ /{\rm MHz}\)  and \(\partial \alpha /\partial \nu \approx 0.8^\circ /{\rm MHz}\)  as \(L\)  is varied between 6 and 20. The magnitude of the slope decreases for larger \(L\)  and must asymptote to zero as \(L\)  tends to infinity.
<FIGURE>